In this paper, we will work in the limit of instantaneous reheating.
This facilitates our analysis; a more rigorous treatment would require a lattice simulation [1]}, [2]}, [3]} (see also Ref. [4]}).
It is moreover justified by the fact that reheating proceeds very fast after axion inflation, if a large amount of energy is already transferred to the hypercharge sector towards the end of inflation, which then quickly thermalizes in consequence of the SM gauge interactions.
In this case, we are able to employ the fermion symmetries in Eq. (REF ) as initial conditions for our description of the radiation-dominated era.
Similarly, the temperature of the SM plasma shortly after the end of inflation coincides with the reheating temperature in the limit of instantaneous reheating,
\(T_{\rm end} \simeq T_{\rm reh} \simeq \left(\frac{90}{\pi ^2 g_*}\right)^{1/4} \sqrt{H_{\rm end} M_{\rm Pl}} \,.\) 
